Abstract:
Hilbert space operators have been studied by many mathematicians. These
operators are of great importance since they are useful in formulation of
principles of mathematical analysis and quantum mechanics. The operators
include normal operators, posinormal operators, hyponormal operators,
normaloid operators among others. Certain properties of posinormal
operators have been characterized like continuity and linearity but numerical
ranges and spectra of posinormal operators have not been considered.
Also the relationship between the numerical range and spectrum has not
been determined for posinormal operators. The objectives of this study
have been: to investigate numerical ranges of posinormal operators, to
investigate the spectra of posinormal operators and to establish the relationship
between the numerical range and spectrum of a posinormal operator.
The methodology involved use of known inequalities like Cauchy-
Schwartz inequality and the polarization identity to determine the numerical
range and spectrum of posinormal operators and our technical
approach involved use of tensor products. We have shown that the numerical
range of a posinormal operator A is nonempty, contains zero and
is an ellipse whose foci are the eigenvalues of A. We have also proved that
the spectrum of a bounded posinormal operator A acting on a complex
Hilbert space H satisfies Xia’s property; and doubly commuting n-tuples
of posinormal operators are jointly normaloid. The results obtained are
applicable in classification of Hilbert space operators and shall be applied
in other fields like quantum information theory to optimize minimal output
entropy of quantum channel; to detect entanglement using positive
maps; and for local distinguishability of unitary operators.
